- #1

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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with another aspect of the proof of Proposition 1.2.17 ... ...

Duistermaat and Kolk's Proposition 1.2.17 and the preceding definition (regarding open and closed sets in a set V) read as follows:

View attachment 7735

View attachment 7736D&K write that the proof of (i) is immediate from the definitions ... but I have been unable to formulate a rigorous proof of (i) ... could someone please demonstrate a rigorous proof of (i) ... ...Hope that someone can help ...

Peter========================================================================================

D&K's definitions and early results on open and closed sets may be helpful to MHB members reading and following the above post ... so I am providing the same ... as follows:View attachment 7737

View attachment 7738Hope that helps ... ...

Peter

I am focused on Chapter 1: Continuity ... ...

I need help with another aspect of the proof of Proposition 1.2.17 ... ...

Duistermaat and Kolk's Proposition 1.2.17 and the preceding definition (regarding open and closed sets in a set V) read as follows:

View attachment 7735

View attachment 7736D&K write that the proof of (i) is immediate from the definitions ... but I have been unable to formulate a rigorous proof of (i) ... could someone please demonstrate a rigorous proof of (i) ... ...Hope that someone can help ...

Peter========================================================================================

D&K's definitions and early results on open and closed sets may be helpful to MHB members reading and following the above post ... so I am providing the same ... as follows:View attachment 7737

View attachment 7738Hope that helps ... ...

Peter

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